The Role of Sine and Cosine Components in Love Waves and Rayleigh Waves for Energy Hauling During Earthquake

The Role of Sine and Cosine Components in Love Waves and Rayleigh Waves for Energy Hauling during Earthquake

Zainal Abdul Aziz and Dennis Ling Chuan Ching Department of Mathematics, Faculty of Science, University Technology Malaysia, 81310 Johor, Malaysia

Abstract: Problem statement: Love waves or shear waves are referred to dispersive SH waves. The diffusive parameters are found to produce complex amplitude and diffusive waves for energy hauling. Approach: Analytical solutions are found for analysis and discussions. Results: When the standing waves are compared with the Love waves, similar cosine component is found in both waves. Energy hauling between waves is impossible for Love waves if the cosine component is independent. In the view of 3D particle motion, the sine component of the Rayleigh waves will interfere with shear waves and thus the energy hauling capability of shear waves are formed for generating directed dislocations. Nevertheless, the cnoidal waves with low elliptic parameter are shown to act analogous to Rayleigh waves. Conclusion: Love waves or Rayleigh waves are unable to propagate energy once the sine component does not exist.

Key words: Energy hauling, sine and cosine, amplitude, love waves, Rayleigh waves, Cnoidal waves, earthquake, phase velocity, periodic, isotropic medium, illustrated, anisotropic

INTRODUCTION seismology, Love waves are referred to shear waves and SH waves in layered medium are similar to Love waves. For this reason, the similarity between standing Seismic body waves are classified to P, SH and SV waves. Similar waves have being derived in vector waves and Love waves must be exploited. Complexity form (Ari-Menahem and Singh, 1981). These waves of fractures and Love waves must be explained to will give the surface Rayleigh waves and Loves waves resolve the mechanism of vein structures formation that decay exponentially with depth. Rayleigh waves especially the direction of dislocation. and Love waves are recognized for near surface Standing waves and progressive waves are displacement and there are categorized under dispersive different in general. Standing waves do not involve in waves since the phase and group velocities are different the form of energy and momentum transfer between (Bhatnagar, 1979). Tsunami waves or long waves are waves separated by nodes while progressive waves are also the surface waves. However, these waves are propagating the energy. Since Rayleigh waves are near dissimilar with Rayleigh waves or Love waves (Ribal, surface waves, multi layer mediums are not suitable for 2007; 2008). showing near surface displacement. However, the Standing waves are known as stationary waves that splitting shear waves or SV waves from Rayleigh do not propagate. There arise in a motionless medium waves have affected the Rayleigh wave’s dispersive as a result of interference involving two waves traveling behavior for layered anisotropic medium (Zhang et al ., in opposite directions. Similar incidence can be 2009). The statement for splitting SV waves is explained through the movement along the tectonic progressive in layered medium must be explained. plate where the interference of waves is invincible. The Energy hauling capability should be considered as well. capability of standing waves to propagate forward has Sine and cosine components in progressive waves attracted researchers’ interest. In experimental study, and standing waves explain the different between (Brothers et al ., 1996) attributed the standing waves Rayleigh and Love waves. Seismic waves are periodic from systematic fractures such as vein structures during waves and thus the displacement has a periodic seismic wave shaking. Later, Tsuneo and Yujiro variation with time and distance. Despite being (Ohsumi and Ogawa, 2008) found vein structures that periodic, conidial waves have sharper crest and flatter are formed by shear waves during earthquake. In troughs than sinusoidal waves. Usually, these waves are Corresponding Author: Zainal Abdul Aziz, Department of Mathematics, Faculty of Science, University Technology Malaysia, 81310 Johor, Malaysia 1550 Am. J. Applied Sci., 7 (12): 1550-1557, 2010 referred to ocean waves which are distinguished by the Where: elliptic parameter. When the elliptic parameter reduces C = The velocity of the phase to zero, smooth repetitive oscillations are formed and δ = α or β = The velocity of the P or S waves these waves are similar to seismic waves. Both waves respectively should look alike at medium surface if the antinodes of K = Refers to the wavenumbers the waves are ignored. Eventually, it must be explained η η η the Rayleigh wave has a possibility to transform into δ = α or β = Known as diffusion parameter Cnoidal waves once the elliptic parameter is increasing to produce nonlinear seismic waves that can propagate Inserting the relations (4) and (5) into equations energy. Apparently, nonlinear waves are known for self (1), (2) and (3), we will get the dispersive form of reinforcing and capable of propagating vast energy in seismic waves as: the medium (Dauxious and Peyrard, 2006; Drazin and

Johnson, 1989). U=± Aa( na) expikct[ ( −η x∓ z )] (6) P xα z α MATERIALS AND METHODS

U= Ba( ∓ na) expikct[ ( − x ∓ η z )] (7) Problem statement: Seismic waves are divided into SV zβ x β two groups P and S in according to the propagation direction. The S waves again are categorized into two U= Ca exp[ ik( ct − x∓ η β z )] (8) sub groups SV and SH waves. The explanations for SH y seismic waves in energy hauling and 3D particle motion are more remarkable if these waves are featured in sine The combination of Eq. 6 and 7 will form the and cosine components. Rayleigh waves while Eq. 8 alone is known as Love The combination of the Loves waves and Rayleigh waves (Ben-Menahem and Singh, 1981). These dispersive waves will give the 3D particle motion as shown in waves will decay exponentially with finite depth. Fig. 1. P, SV and SH waves introduced by Ben- For the case of δ>c, we get the complex ηδ. Menahem and Singh (1981) read: Inserting the complex ηδ to either Eq. 6-8, the selected

+ Eq. 8 reads: =() + − lx nz  UP A la x na z exp iw t   (1) α 

U= Cexp( i ηβ z) exp[] ik() ct − x (9) lx+ nz  SH U=−+ B() na la exp iw t − (2) SV x z β     After linear polarization with respect to y-direction. C lx+ nz  Is the amplitude while exp(i ηβz) is referring to the U= Ca expiw t − (3) SH y β   diffusive term. Apparently, Eq. 9 is non diffusive since   the term exp(i ηβz) is in complex (Bhatnagar, 1979). Where: l and n = The vector components Refer to the unit vector ax, a y and a z = α and β = The velocity for P and S waves A, B and C = Refer to the amplitude of the waves with finite depth z and w = The frequency of the waves

For elliptic particle motion, similar frequency with respect to the wave is expected. It is appropriate to reduce (1), (2) and (3) through the relations that read:

c2 δ nδ = − 1, c = (4) δ2 l

c2  x± − 1  z 2  lx+ nz x + nzl δ  = = (5) δ δ c l Fig. 1: The seismic wave’s framework 1551 Am. J. Applied Sci., 7 (12): 1550-1557, 2010

components introduced by Ben-Menahem and Singh (1981), the stress read:

τ=µu =µ[ Cexp( −η i kz) + Cexpi( η kz )] 32 2,311β 2 β (11) exp[] ik() ct− x

τ=µu =µ expikct[ ( − x )] (12) 32 2,3 1

where, u 1, u2, u 3 are in x, y, z directions. Equation 11 shows the stress within the layered medium while the Fig. 2: The amplitude and diffusion behavior of the S free surface stress is depicted through Eq. 12 when z = 0. waves Despite the absent of complex amplitude and diffusive term, the periodic sinusoidal waves are still exist. Love waves are usually explained in a layer over a half space medium. There is significant different between free surface stress and τ stress. Stress is also 32 recognized as diffusion energy. Free surface has less energy at the free surface whereas τ32 stress produces radical stress in close proximity to the surface featuring complex or non amplitude as illustrated in Fig. 4a and 4b respectively. Equation 10 can be written in cosine form with amplitude C = C that reads: 1 2

Fig. 3: The interference of S waves U= Ca cos( η kh) exp[] ik() ct − x (13) SH y β

From Eq. 6-8, ηα and ηβ give the dispersive waves

with amplitude A, B and C. The explanation can go into Equation 13 shares the similarity with the standing diffusive characteristic of the waves and also the waves (Bhatnagar, 1979). complex amplitudes. According to (Bhatnagar, 1979), the standing Complex amplitude is usually referred to phasor or waves do not include in the form of energy and amplitude of the waves. Figure 2 illustrates the momentum transfer between waves separated by nodes. propagation behavior of the SV waves featuring This explains Love waves do not forward energy. Since complex amplitude. It is possible for SV waves become Love waves only contribute to the cosine component, SH waves after polarization. Conforming to the rotating this kind of waves are considered as non moving waves behavior, S wave is plotted for the dispersion and which vibrate at the origin without participating in diffusion behaviors as illustrated in Fig. 3. S waves energy transformation. Conforming to the definition of have dispersed into many and diffused with depth z. standing waves, the capability of Love waves in energy These S waves are not featuring energy hauling behavior. hauling is non-existence. Fig. 3: illustrates the In the following, the justifications will be shown. intersection path of the Love waves featuring the

Love waves cosine component: For finite layer of interference of the wave moving in opposite depth, the Love waves are referred to the Eq. 8. direction to diminish the energy and momentum According to Fig. 1 the waves move in both positive transfer between nodes. and negative y direction as: The stress component between 2 layers can be written as: =[ ( −η+) ( η)] []() − USH a y C 1 exp ikβ z C 2 exp ik β z exp ik ct x (10) C()− exp()() −η i kh + expi η kh = 1 1 1 ()η µ (14) The induced stress by the propagating Love waves 2 2 C()− exp()() −η i kh + exp i η kh ()η µ 2 2 2 in the medium cannot be ignored. By using the stress 1 1 1552 Am. J. Applied Sci., 7 (12): 1550-1557, 2010

where the shear modulus of waves are related to shallowness’ parameter such that the shear stress are proportional to the shallow of the layered medium. Letting A = C B = C et a 1 = η and et a 2 = η in 1, 2 1 2 Fig. 5, the complex diffusive parameter are examined for the energy diffusion and Love wave dispersion in two layered medium. It is observed that the displacement will be greater at layer 2 according to the depth of the stress distributed by Love waves. The amplitudes of the Love waves in two different layers of medium are seen to generate nearly equal stress at both directions when both the amplitudes C 1 and C 2 are nearly similar. In fact, the medium density for two layers of medium must be different. In the same way Eq. 14 can transform into cosine form that reads:

(a) Acos(η kh) exp[ ik( ct − x )] = 1 (15) ()()η[] − Bcos2 kh exp ik ct x

In the following, the sine and cosine components are studied to show the role of sine and also the cosine components.

Rayleigh waves sine and cosine components: Rayleigh waves and Love waves should not share the same diffusion given that Rayleigh waves are guided near surface waves while Love waves are guided shear waves in a layered isotropic medium. Free surface stress should not consider the shallowness of the medium such that a single layered isotropic medium is discussed for Rayleigh waves instead of two layered medium.

(b) In previous following, Love waves propagate as standing waves since the cosine component exists in SH Fig. 4: (a) The interference of the S waves limiting the waves. However, the Love waves which are also the energy from being brought forward. (b) The S shear waves either SV or SH waves are progressive if waves with high diffusion energy at the there are independent. In the view of 3D particle motion, epicenter. (b) The overlapped amplitude or these shear waves will interfere with the P waves. diffusion energy near the medium surface For the case mentioned by Zhang et al . (2009), the

vein structure or the displacement is brought by the It is recognized in isotropic medium, there exists two kinds of waves: pressure waves and shear waves. shear waves that propagate within the medium. Hence, The two diffusion parameters in 2 layers of isotropic it is well said the P-SV waves which are also the medium are expected to give two kinds of shear waves. Rayleigh waves must show some characteristic of S From Eq. 14, µ and µ are the shear modulus or waves. It can be shown that the SV waves of the 1 2 modulus of rigidity of the medium. When this is applied Rayleigh waves which are also the shear waves are to Love waves, the shear modulus and the density ρ generating stress to the medium and the stress is give the velocity of shear wave Vs that reads: brought forward by another part of the Rayleigh waves or P waves. In other words, the Rayleigh waves involve = µ in the form of energy and momentum transfer between Vs ρ waves separated by nodes. 1553 Am. J. Applied Sci., 7 (12): 1550-1557, 2010

In this study, energy is said being forwarded by the c when ≃0.92 ≃ 0.9591 . The dispersive stress by sine component of the Rayleigh waves from the view of β 3D particle motion by the elliptic waves. These elliptic dispersive SV waves are supposed similar to SH waves seismic waves are built up by P, SV and SH waves as as illustrated in Fig. 5 since both the SH and SV waves illustrated in Fig. 1. The explanation can go into are different in propagation direction only. When SV diffusion energy or stress when the Rayleigh waves are waves are interfered with P waves, Rayleigh waves or proven for carrying sine and cosine components. P-SV waves are formed. It is shown that Rayleigh Rayleigh waves are built from P and SV waves waves are dispersive when the phase velocity and the from (6) and (7) that reads: velocity of the wave are different or c2 ≤ δ2. Dispersion splitting as mentioned by Zhang et al . (2009) will Aa()()+η iα a exp −η α kz +  U+ U =x z exp[] ik() ct − x (16) happened when there are two or more travelling SV P SV ()()+η −η  Baz iβ a x exp β kz  waves in different velocities. With the significant different of wave’s velocities in two different layers of Ben-Menahem and Singh (1981) introduces the mediums, the dispersion splitting is expected. stress components for P and S waves. The P waves The integration of stress components (17) and (18) stress components read: into the amplitude of Rayleigh wave is important to show the existence of sine and cosine components. The τ =µ τ =λµ +λ+µ( ) 312u, 1,3 33 1,1 2u 3,3 reconstructed stress embedded P waves and SV waves are shown by Eq. 20 and 21: While the S waves stress component read: 1 c 2   −2 − exp() −γ kz τ=µ( +) τ=µ 2  α  31u 1,3 u, 3,1 33 2 3,3 =()[]() − γ β  UP Biexp ik ct x α  (20) +γ −γ  ()()βexp β kz  where, u1 = u x, u3 = u z. For x and y directions, the stress components after algebraic manipulation read: 1 c 2   −2 −() γ α exp 2 2   c  U=() Bexpikct[]() − x γα  β  γ = − SV   (21) 2α A Bi 2  (17) β2  ()−γ +() −γ  αkz exp β kz 

c2  −γ= − 2 2Bβ Ai 2 2  (18) 1 c  β  Letting Q=( Bexpik[ ( ct − x )]) , W= − 2 −  γ β 2  α From (17) and (18), the determinant reads: into (20) and (21), we get:

2 2 2 2 c  UP  U SV  2− −γγ= 4α β 0 (19) + = 1 (22) β2       Q  Q 

2 2 γ = − c δ α β ψ = c Once Eq. 22 is compared to ellipse equation: where, δ 1 2 and = , . Let 2 δ β α2 2( −+) 2 ( −=) and = 3 , Eq. 19 becomes: sin wt kx cos wt kx 1 β2

elliptic waves are presented for Rayleigh waves. 3ψ−3 24 ψ+ 2 56 ψ− 320 = Apparently, the sine and cosine form of P waves and SV waves read: ψ= −2 ≈ ψ ψ = + 2 with roots 1 2 0.92 , 2 = 4 and 3 2 . 3 3 U= QU(z)sin(wt − kx) (23) For dispersive waves with phase velocity slower than P the wave’s velocities, only the root ψ1 ≈ 0.92 is valid or 2 2 U= QK(z)cos(wt − kx),K( z) =γ W( z ) (24) c ≤δ . Eventually, the Rayleigh waves are dispersive SV α 1554 Am. J. Applied Sci., 7 (12): 1550-1557, 2010

(a) (b)

Fig. 5: The stress by Love waves in between two layers of isotropic mediums. Amplitudes C1 and C2 are generated by Love waves in Layer 1 and Layer 2 respectively. The diffusive parameters η1 and η2 are referred to the Love waves in two different layers of medium

Equation 23 and 24 are the periodic waves. Rayleigh waves and also the propagation potential. Equation 24 holds cosine component and so the Since both the SV and SH waves have cosine similarity of SV waves and SH waves holding cosine component, the hauling characteristics should not exist. components are shown. Love waves and SV waves are Rayleigh wave capable of bringing diffusion energy recognized to hold cosine component while Rayleigh during earthquake is hence blamed to the sine waves hold sine and also cosine components. Both sine component of P waves. The interference of P waves and cosine components represent different parts of the from the Rayleigh waves with the Love waves is 1555 Am. J. Applied Sci., 7 (12): 1550-1557, 2010 suggested to carry some of the energy forward as well. Apparently, P and SV waves interfered with SH waves build to 3D elliptic waves when the frequencies allowed. Besides the sine component of P waves, there is another possibility for explaining the energy hauling attribute of Rayleigh waves and Love waves. Rayleigh wave is suggested for being conidial waves that do not carry elliptic parameters. Both the waves should hold energy since the stress is adherent. At the following, the energy equation by Bhatnagar (1979) is exploited to justify the hypothesis. The energy equation reads:

12 1 Uξ =() − f() u (25) 2 6k Fig. 6: The root of the energy equation

Where:

( ) =( −γ)( −β)( α− ) fuue u e e u (26)

The root of the energy Eq. 26 is in according to Fig. 6. The energy Eq. 25 can be written as:

du d ξ = (27) 1 1 ()()()u−γ u −β α− u  2()3K 2 e e e  Fig. 7: The transformation of Rayleigh waves to conidial waves with the increment of elliptic p2 α − β parameter, s from 0 to 200 with time t. Letting p2 = α -u, α −β = , s2 = e ee , e e e 2 α − γ q e e Referring to Fig. 7 it is evidenced that the troughs pα − u q = = e into (27), we get: of the waves have changed when the elliptic parameter α − β α − β e e e e is increasing. Without the elliptic parameters, the conical wave is simply a sinusoidal wave or acoustic wave that is similar to Rayleigh wave. In fact, these ξ = 12Kq d q ∫0 1 2 nonlinear waves are capable of carrying and α − γ 2 2 2 e e ()()1− sq 1 − q    (28) transporting vast energy.

12K − = Sn1 () q,s α − γ RESULTS AND DISCUSSION e e

The cosine component explained the capability of Or: shear waves carrying energy. Love waves which are similar to standing waves do not carry energy forward α − γ  once the standing waves are defined as the interference ξ=β+α−β2 ξ e e u()() Cn ,s  (29) of two waves moving in opposite direction. High 12K  diffusion energy is expected at the rupturing path with the energy diffused when they arrive at medium Equation 29 is the Cnoidal waves extracted from surface. However, the near surface stress and the stress energy equation where Sn and Cn are the Jacobian by interfered Love waves near to the surface might Elliptic functions. s is the elliptic parameter to decide differ given that the diffusive terms are considered for the troughs of the Cnoidal waves. the stress component of the interfered waves. 1556 Am. J. Applied Sci., 7 (12): 1550-1557, 2010

Overlapped amplitude is shown by interfered waves. Brothers, R.J., A.E.S. Kemp and A.J. Maltman, 1996. From the discussions, the shear waves either SV or SH Mechanical development of vein structures due to waves have tendency for being interfered by the P the passage of earthquake waves through poorly waves which are part of the Rayleigh waves. Rayleigh consolidated sediments. Tectonophysics, 260: 227-244. waves only consist of sine and cosine component. If the DOI: 10.1016/0040-1951(96)00088-1 cosine component is proven for unable to carry energy Dauxious, T. and M. Peyrard, 2006. Physics of forward, then the sine component is responsible for Solitons. 1st Edn., Cambridge University Press, propagating the energy. The vein structure found after United Kingdom, pp: 422. an earthquake illustrates the shear waves or SV waves http://www.amazon.com/gp/search?index=books&l can propagate in the isotropic medium. Through 3D inkCode=qs&keywords=0521854210 particle motion, it is suggested that these SV waves and Drazin, P.G. and R.S. Johnson, 1989. Solitons: An Love waves are brought forward by the P waves which introduction. 1st Edn., Cambridge University Press, are belong to Rayleigh waves. There is another United Kingdom, pp: 226. ISBN: 0-521-33655-4 possibility for energy hauling during earthquake. The Gicev, V. and M.D. Trifunac, 2010. Amplification of Cnoidal waves which are also known as nonlinear linear strain in a layer excited by a shear wave periodic waves are differed by elliptic parameter. When earthquake pulse. Soil Dyn. Earthq. Eng., the elliptic parameter is too small, the Cnoidal waves 30: 1073-1081. DOI: should look alike seismic waves for which the self 10.1016/J.SOILDYN.2010.04.018 reinforced capability of these nonlinear waves Ohsumi, T. and Y. Ogawa, 2008. Vein structures, like propagate energy during an earthquake. ripple marks, are formed by short wavelength shear Recently, the amplification of linear strain in a waves. J. Struct. Geol., 30: 719-724. DOI: layer excited by a shear wave earthquake pulses are 10.1016/J.JSG.2008.02.002 found by Gicev and Trifunac (2010). The interference Ribal, A., 2007. Numerical study on the effect of linear of two waves in opposite directions is exploited and the friction of long-wave propagation and breaking amplification is due to sine pulse. The simulated figures with sloping bottom. J. Math. Stat., 3: 263-267. by Gicev and Trifunac (2010) illustrate flat troughs DOI: 10.3844/.2007.263.267 similar to conical waves as well. Hence, it is well said Ribal, A., 2008. Numerical study of tsunami waves the sine component of Rayleigh waves are propagating with sloping bottom and nonlinear friction. J. the energy of interfered Love waves. Ability of Math. Stat., 4: 41-45. DOI: 10.3844/.2008.41.45 Conidial waves interfering the shear waves are Zhang, S.X., Y. Wang, H.W. Zhou and L.S. Chan, suggested for further study. 2009. Dispersion splitting of Rayleigh waves in layered azumuthally anisotropic media. J. Applied CONCLUSION Geophys., 67: 130-142. DOI: 10.1016/J.JAPPGEO.2008.10.008 Sine components of Rayleigh waves are responsible for propagating the cosine components. The energy or stress relating to the cosine component does not carry the energy forward. There is another likelihood for explaining the incident of energy hauling during an earthquake. Cnoidal waves with low elliptic parameters are similar to Rayleigh waves and the self reinforced capabilities of these nonlinear waves are known for carrying energy.

REFERENCES

Ari-Menahem, B. and S.J. Singh, 1981. Seismic Waves and Sources. 2nd Edn., Springer-Verlag, New York, pp: 1136. ISBN: 0-486-40461-7 Bhatnagar, P.L., 1979. Nonlinear Waves in One Dimensional Dispersive System. 1st Edn., Thomson Press Limited, India, pp: 142. ISBN: 0198535317 1557